Structural classification of continuous time Markov chains with applications

Abstract

This paper is motivated by demands in stochastic reaction network theory. The Q-matrix of a stochastic reaction network can be derived from the reaction graph, an edgelabelled directed graph encoding the jump vectors of an associated continuous time Markov chain on an ambient space N d0 . An open question is whether the ambient space N d 0 can be decomposed into neutral, trapping, and escaping states, and open and closed communicating classes, based on the reaction graph alone. We characterize the structure of the state space of a Q-matrix on N d 0 that generates continuous time Markov chains taking values in N d 0 , in terms of the set of jump vectors and their corresponding transition rate functions. We also define structural equivalence of two Q-matrices, and provide sufficient conditions for structural equivalence. Such stochastic processes are abundant in applications. We apply our results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.

Full classification of dynamics for one-dimensional continuous time Markov chains with polynomial transition rates

“…for k ∈ N and x ∈ N 0 , where x j = j−1 i=0 (x − i) is descending factorial. By (H4), (A1)-(A2) are satisfied; moreover, the irreducibility of Y t also follows from [37]. Under (H5), the mass-action kinetics yield (A3)-(A4).…”

“…By Theorem 3.4, the first is positive recurrent and admits an exponentially ergodic stationary distribution on N 0 since α = −κ 2 and R = 1, while by Theorem 3.1, the second reaction network is explosive for any initial state since α = κ 3 > 0 and R = 2. Indeed, these two reaction networks are structurally equivalent in that there is only one irreducible component N 0 [37].…”

“…The threshold parameters are the same as in (i). Let ∂ = {0} with ∂ c = N an irreducible component [37]. By Theorems 3.4, the network has a uniformly exponentially ergodic QSD on N.…”

“…for the set of consecutive integers from m to n, with m, n ∈ N 0 ∪ {+∞}. The notation is adopted from [37].…”